[Variétés hyperboliques, produits amalgamés et exposants critiques]
Nous donnons une preuve nouvelle d'un résultat dû à Y. Shalom ; précisément, nous montrons que, si le groupe fondamental d'une variété hyperbolique réelle compacte de dimension n est le produit libre de ses sous-groupes A et B amalgamé sur C, alors l'exposant critique de C est plus grand que n−2. La preuve, géométrique, permet de traiter le cas d'égalité ainsi qu'une extension au cas de courbure variable.
We give a new proof of a result due to Y. Shalom: if the fundamental group of a compact real hyperbolic manifold of dimn is a free product of its subgroups A and B over the amalgamated subgroup C, then the critical exponent of C is not smaller than n−2. The proof, which is geometric, allows one to treat the equality case and an extension to variable curvature.
Accepté le :
Publié le :
@article{CRMATH_2003__336_3_257_0,
author = {Besson, G\'erard and Courtois, Gilles and Gallot, Sylvestre},
title = {Hyperbolic manifolds, amalgamated products and critical exponents},
journal = {Comptes Rendus. Math\'ematique},
pages = {257--261},
publisher = {Elsevier},
volume = {336},
number = {3},
year = {2003},
doi = {10.1016/S1631-073X(02)00019-5},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)00019-5/}
}
TY - JOUR AU - Besson, Gérard AU - Courtois, Gilles AU - Gallot, Sylvestre TI - Hyperbolic manifolds, amalgamated products and critical exponents JO - Comptes Rendus. Mathématique PY - 2003 SP - 257 EP - 261 VL - 336 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)00019-5/ DO - 10.1016/S1631-073X(02)00019-5 LA - en ID - CRMATH_2003__336_3_257_0 ER -
%0 Journal Article %A Besson, Gérard %A Courtois, Gilles %A Gallot, Sylvestre %T Hyperbolic manifolds, amalgamated products and critical exponents %J Comptes Rendus. Mathématique %D 2003 %P 257-261 %V 336 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)00019-5/ %R 10.1016/S1631-073X(02)00019-5 %G en %F CRMATH_2003__336_3_257_0
Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre. Hyperbolic manifolds, amalgamated products and critical exponents. Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 257-261. doi : 10.1016/S1631-073X(02)00019-5. http://archive.numdam.org/articles/10.1016/S1631-073X(02)00019-5/
[1] Lemme de Schwarz réel et applications géométriques, Acta Math., Volume 183 (1999), pp. 145-169
[2] G. Carron, E. Pedon, On the differential form spectrum of hyperbolic manifold, to appear
[3] Filling Riemannian manifolds, J. Differential Geom., Volume 18 (1983), pp. 1-147
[4] Hyperbolic Manifolds and Discrete Groups, Progress in Math., 183, Birkhaüser, 2001
[5] Free quotients and the first Betti number of some hyperbolic manifolds, Transformation Groups, Volume 1 (1996) no. 1, 2, pp. 71-82
[6] Arbres, amalgames, SL2, Astérisque, Volume 46 (1977)
[7] Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math., Volume 152 (2000), pp. 113-182
[8] W. Thurston, The geometry and topology of three-manifolds, Lecture Notes, Princeton
Cité par Sources :
